A matheuristic approach for solving a home health care problem

We deal with a Home Health Care Problem (HHCP) which objective consists in constructing the optimal routes and rosters for the health care staffs. The challenge lies in combining aspects of vehicle routing and staff rostering which are two well known hard combinatorial optimization problems. To solve this problem, we initially propose an integer linear programming formulation (ILP) and we tested this model on small instances. To deal with larger instances we develop a matheuristic based on the decomposition of the ILP formulation into two problems. The first one is a set partitioning like problem and it represents the rostering part. The second problem consists in the routing part. This latter is equivalent to a Multi-depot Traveling Salesman Problem with Time Windows (MTSPTW).     

The Solution of Massive Generalized Set Partitioning Problems in Aircrew Rostering

The scheduling and rostering of personnel is a problem that occurs in many organizations. Aircrew scheduling has attracted considerable attention with many heuristic methods being proposed, but in recent times set partitioning optimization methods have become more popular. The aircrew rostering problem is discussed and formulated as a generalized set partitioning model. Because of the extremely large optimization models that are generated in practical situations, some special computational techniques have been developed to produce solutions efficiently. These techniques are used to solve problems arising from an airline application in which set partitioning models with more than 650 constraints and 200 000 binary variables are generated. The solutions are produced on a Motorola 68020 microprocessor in little more than three hours.     

A variable neighborhood search based matheuristic for nurse rostering problems

A practical nurse rostering problem, which arises at a ward of an Italian private hospital, is considered. In this problem, it is required each month to assign shifts to the nursing staff subject to various requirements. A matheuristic approach is introduced, based on a set of neighborhoods iteratively searched by a commercial integer programming solver within a defined global time limit, relying on a starting solution generated by the solver running on the general integer programming formulation of the problem. Generally speaking, a matheuristic algorithm is a heuristic algorithm that uses non trivial optimization and mathematical programming tools to explore the solutions space with the aim of analyzing large scale neighborhoods. Randomly generated instances, based on the considered nurse rostering problem, were solved and solutions computed by the proposed procedure are compared to the solutions achieved by pure solvers within the same time limit. The results show that the proposed solution approach outperforms the solvers in terms of solution quality. The proposed approach has also been tested on the well known Nurse Rostering Competition instances where several new best results were reached.     

Elastic Constraint Branching, the Wedelin/Carmen Lagrangian Heuristic and Integer Programming for Personnel Scheduling

The Wedelin algorithm is a Lagrangian based heuristic that is being successfully used by Carmen Systems to solve large crew pairing problems within the airline industry. We extend the Wedelin approach by developing an implementation for personnel scheduling problems (also termed staff rostering problems) that exploits the special structure of these problems. We also introduce elastic constraint branching with the twin aims of improving the performance of our new approach and making it more column generation friendly. Numerical results show that our approach can outperform the commercial solver CPLEX on difficult commercial rostering problems.     

A Neural Network Application in Personnel Scheduling

In this paper, we present and evaluate a neural network model for solving a typical personnel-scheduling problem, i.e. an airport ground staff rostering problem. Personnel scheduling problems are widely found in servicing and manufacturing industries. The inherent complexity of personnel scheduling problems has normally resulted in the development of integer programming-based models and various heuristic solution procedures. The neural network approach has been admitted as a promising alternative to solving a variety of combinatorial optimization problems. While few works relate neural network to applications of personnel scheduling problems, there is great theoretical and practical value in exploring the potential of this area. In this paper, we introduce a neural network model following a relatively new modeling approach to solve a real rostering case. We show how to convert a mixed integer programming formulation to a neural network model. We also provide the experiment results comparing the neural network method with three popular heuristics, i.e. simulated annealing, Tabu search and genetic algorithm. The computational study reveals some potential of neural networks in solving personnel scheduling problems.     

A dual ascent procedure for the set partitioning problem

The set partitioning problem is a fundamental model for many important real-life transportation problems, including airline crew and bus driver scheduling and vehicle routing.In this paper we propose a new dual ascent heuristic and an exact method for the set partitioning problem. The dual ascent heuristic finds an effective dual solution of the linear relaxation of the set partitioning problem and it is faster than traditional simplex based methods. Moreover, we show that the lower bound achieved dominates the one achieved by the classic Lagrangean relaxation of the set partitioning constraints. We describe a simple exact method that uses the dual solution to define a sequence of reduced set partitioning problems that are solved by a general purpose integer programming solver. Our computational results indicate that the new bounding procedure is fast and produces very good dual solutions. Moreover, the exact method proposed is easy to implement and it is competitive with the best branch and cut algorithms published in the literature so far.     

A generic approach to proving NP-hardness of partition type problems

This note presents a generic approach to proving NP-hardness of unconstrained partition type problems, namely partitioning a given set of entities into several subsets such that a certain objective function of the partition is optimized. The idea is to represent the objective function of the problem as a function of aggregate variables, whose optimum is achieved only at the points where problem Partition (if proving ordinary NP-hardness), or problem 3-Partition or Product Partition (if proving strong NP-hardness) has a solution. The approach is demonstrated on a number of discrete optimization and scheduling problems.     

An estimation of distribution algorithm with intelligent local search for rule-based nurse rostering

This paper proposes a new memetic evolutionary algorithm to achieve explicit learning in rule-based nurse rostering, which involves applying a set of heuristic rules for each nurse's assignment. The main framework of the algorithm is an estimation of distribution algorithm, in which an ant-miner methodology improves the individual solutions produced in each generation. Unlike our previous work (where learning is implicit), the learning in the memetic estimation of distribution algorithm is explicit, that is, we are able to identify building blocks directly. The overall approach learns by building a probabilistic model, that is, an estimation of the probability distribution of individual nurse–rule pairs that are used to construct schedules. The local search processor (ie the ant-miner) reinforces nurse–rule pairs that receive higher rewards. A challenging real-world nurse rostering problem is used as the test problem. Computational results show that the proposed approach outperforms most existing approaches. It is suggested that the learning methodologies suggested in this paper may be applied to other scheduling problems where schedules are built systematically according to specific rules.     

Branching in branch-and-price: a generic scheme

Developing a branching scheme that is compatible with the column generation procedure can be challenging. Application specific and generic schemes have been proposed in the literature, but they have their drawbacks. One generic scheme is to implement standard branching in the space of the compact formulation to which the Dantzig-Wolfe reformulation was applied. However, in the presence of multiple identical subsystems, the mapping to the original variable space typically induces symmetries. An alternative, in an application specific context, can be to expand the compact formulation to offer a wider choice of branching variables. Other existing generic schemes for use in branch-and-price imply modifications to the pricing problem. This is a concern because the pricing oracle on which the method relies might become obsolete beyond the root node. This paper presents a generic branching scheme in which the pricing oracle of the root node remains of use after branching (assuming that the pricing oracle can handle bounds on the subproblem variables). The scheme does not require the use of an extended formulation of the original problem. It proceeds by recursively partitioning the subproblem solution set. Branching constraints are enforced in the pricing problem instead of being dualized via Lagrangian relaxation, and the pricing problem is solved by a limited number of calls to the pricing oracle. This generic scheme builds on previously proposed approaches and unifies them. We illustrate its use on the cutting stock and bin packing problems. This is the first branch-and-price algorithm capable of solving such problems to integrality without modifying the subproblem or expanding its variable space.     

Selecting and weighting features using a genetic algorithm in a case-based reasoning approach to personnel rostering

Personnel rostering problems are highly constrained resource allocation problems. Human rostering experts have many years of experience in making rostering decisions which reflect their individual goals and objectives. We present a novel method for capturing nurse rostering decisions and adapting them to solve new problems using the Case-Based Reasoning (CBR) paradigm. This method stores examples of previously encountered constraint violations and the operations that were used to repair them. The violations are represented as vectors of feature values. We investigate the problem of selecting and weighting features so as to improve the performance of the case-based reasoning approach. A genetic algorithm is developed for off-line feature selection and weighting using the complex data types needed to represent real-world nurse rostering problems. This approach significantly improves the accuracy of the CBR method and reduces the number of features that need to be stored for each problem. The relative importance of different features is also determined, providing an insight into the nature of expert decision making in personnel rostering.     

Solving a nurse scheduling problem with knapsacks,networks and tabu search

This paper illustrates how a modern heuristic and two classical integer programming models have been combined to provide a solution to a nurse rostering problem at a major UK hospital. Neither a heuristic nor an exact approach based on a standard IP package was able to meet all the practical requirements. This was overcome by using a variant of tabu search as the core method, but applying knapsack and network flow models in pre- and post-processing phases. The result is a successful software tool that frees senior nursing staff from a time consuming administrative task.     

Two-level decomposition algorithm for crew rostering problems with fair working condition

A typical railway crew scheduling problem consists of two phases: a crew pairing problem to determine a set of crew duties and a crew rostering problem. The crew rostering problem aims to find a set of rosters that forms workforce assignment of crew duties and rest periods satisfying several working regulations. In this paper, we present a two-level decomposition approach to solve railway crew rostering problem with the objective of fair working condition. To reduce computational efforts, the original problem is decomposed into the upper-level master problem and the lower-level subproblem. The subproblem can be further decomposed into several subproblems for each roster. These problems are iteratively solved by incorporating cuts into the master problem. We show that the relaxed problem of the master problem can be formulated as a uniform parallel machine scheduling problem to minimize makespan, which is NP-hard. An efficient branch-and-bound algorithm is applied to solve the master problem. Effective valid cuts are developed to reduce feasible search space to tighten the duality gap. Using data provided by the railway company, we demonstrate the effectiveness of the proposed method compared with that of constraint programming techniques for large-scale problems through computational experiments.     

Progress control in iterated local search for nurse rostering

This paper describes an approach in which a local search technique is alternated with a process which ‘jumps’ to another point in the search space. After each ‘jump’ a (time-intensive) local search is used to obtain a new local optimum. The focus of the paper is in monitoring the progress of this technique on a set of real world nurse rostering problems. We propose a model for estimating the quality of this new local optimum. We can then decide whether to end the local search based on the predicted quality. The fact that we avoid searching these bad neighbourhoods enables us to reach better solutions in the same amount of time. We evaluate the approach on five highly constrained problems in nurse rostering. These problems represent complex and challenging real world rostering situations and the approach described here has been developed during a commercial implementation project by ORTEC bv.     

An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts

This paper presents a new exact algorithm for the Capacitated Vehicle Routing Problem (CVRP) based on the set partitioning formulation with additional cuts that correspond to capacity and clique inequalities. The exact algorithm uses a bounding procedure that finds a near optimal dual solution of the LP-relaxation of the resulting mathematical formulation by combining three dual ascent heuristics. The first dual heuristic is based on the q-route relaxation of the set partitioning formulation of the CVRP. The second one combines Lagrangean relaxation, pricing and cut generation. The third attempts to close the duality gap left by the first two procedures using a classical pricing and cut generation technique. The final dual solution is used to generate a reduced problem containing only the routes whose reduced costs are smaller than the gap between an upper bound and the lower bound achieved. The resulting problem is solved by an integer programming solver. Computational results over the main instances from the literature show the effectiveness of the proposed algorithm.      

Multi-phase dynamic constraint aggregation for set partitioning type problems

Dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems. This speed up is mostly due to the use, at each iteration, of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model. This aggregation is updated when needed to ensure the exactness of the overall approach. In this paper, we propose a new version of this method, called the multi-phase dynamic constraint aggregation method, which essentially adds to the original method a partial pricing strategy that involves multiple phases. This strategy helps keeping the size of the aggregated problem as small as possible, yielding a faster average computation time per iteration and fewer iterations. We also establish theoretical results that provide some insights explaining the success of the proposed method. Tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2,000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4.5.     

A reformulation-convexification approach for solving nonconvex quadratic programming problems

In this paper, we consider the class of linearly constrained nonconvex quadratic programming problems, and present a new approach based on a novel Reformulation-Linearization/Convexification Technique. In this approach, a tight linear (or convex) programming relaxation, or outer-approximation to the convex envelope of the objective function over the constrained region, is constructed for the problem by generating new constraints through the process of employing suitable products of constraints and using variable redefinitions. Various such relaxations are considered and analyzed, including ones that retain some useful nonlinear relationships. Efficient solution techniques are then explored for solving these relaxations in order to derive lower and upper bounds on the problem, and appropriate branching/partitioning strategies are used in concert with these bounding techniques to derive a convergent algorithm. Computational results are presented on a set of test problems from the literature to demonstrate the efficiency of the approach. (One of these test problems had not previously been solved to optimality). It is shown that for many problems, the initial relaxation itself produces an optimal solution.     

A dual ascent heuristic for obtaining a lower bound of the generalized set partitioning problem with convexity constraints

In this paper we propose a dual ascent heuristic for solving the linear relaxation of the generalized set partitioning problem with convexity constraints, which often models the master problem of a column generation approach. The generalized set partitioning problem contains at the same time set covering, set packing and set partitioning constraints. The proposed dual ascent heuristic is based on a reformulation and it uses Lagrangian relaxation and subgradient method. It is inspired by the dual ascent procedure already proposed in literature, but it is able to deal with right hand side greater than one, together with under and over coverage. To prove its validity, it has been applied to the minimum sum coloring problem, the multi-activity tour scheduling problem, and some newly generated instances. The reported computational results show the effectiveness of the proposed method.